RESTRICTING THE STEINBERG CHARACTER IN FINITE
SYMPLECTIC GROUPS
JIANBEI AN AND GERHARD HISS
Abstract. We determine the restriction of the Steinberg character of a finite symplectic group of odd characteristic to its maximal parabolic subgroup
stabilizing a line. We relate this restriction to the tensor product of a Weil character with the Steinberg character. As an application we prove that Donovan’s
conjecture has a positive answer for unipotent `-blocks of the six-dimensional
symplectic groups of odd characteristic when ` > 3.
1. Introduction
Parabolic subgroups are “large” subgroups of finite groups of Lie type. Thus
knowledge about the representations of a parabolic subgroup should yield information about the representations of the group itself. Since in general the irreducible
representations of a parabolic subgroup P cannot be classified, one usually restricts
attention to representations of P having its unipotent radical in their kernel.
In classical groups there are distinguished parabolic subgroups, whose Levi subgroups are classical groups of the same type, and whose ordinary irreducible representations can be classified. This is the case, for example, for a finite symplectic
group G = Sp2m (q) and its maximal parabolic subgroup P fixing a line in the
natural representation. It makes sense to ask for the restrictions of the irreducible
characters of G to P .
In this paper we determine the restriction of the Steinberg character of G to P
if q is odd. The parametrization of the irreducible characters of P is rather different
in the two cases of odd and even q. In the latter case, G ∼
= SO2m+1 (q), and this
parametrization resembles the one for the groups SO2m+1 (q) with odd q. Hence
the case of even q is best dealt with in the framework of orthogonal groups.
Assume now that q is odd and let U denote the unipotent radical of P and L its
standard Levi subgroup. Then L is a direct product of a cyclic group of order q − 1
and of L0 = Sp2m−2 (q). The irreducible characters of P with U in their kernels
are in bijection with the irreducible characters of L. The irreducible characters
of P whose kernel is the centre of U , are parametrized by the irreducible characters
of the parabolic subgroup P2m−2 of L0 analogous to P . Finally, there is a fourto-one correspondence between the remaining irreducible characters of P and the
irreducible characters of L0 .
The restriction of the Steinberg character of G to P contains exactly one constituent of the first type, namely the Steinberg character of L. The constituents
of the second type can be computed by restricting the Steinberg character of L0
to P2m−2 . Characters of the third type correspond to the constituents of the product of the Weil character with the Steinberg character of L0 .
1
We explicitly compute all the constituents in the cases G = Sp4 (q) and Sp6 (q). It
turns out that these restrictions are multiplicity free. We use this result to show that
if ` is a prime bigger than 3, all unipotent `-blocks of a given defect occurring in the
groups Sp6 (q) for odd prime powers q not divisible by `, lie in finitely many Morita
equivalence classes. For this purpose we use an approximation to the decomposition
matrix of these groups. Such an approximation is not available for ` = 2 or for
groups Sp2m (q) when m is bigger than 3. We could perhaps extend the above result
to the case ` = 3, but only at the cost of rather tedious technicalities.
We expect that the ideas used in this paper can be extended to other classical
groups to yield qualitative information about decomposition numbers of unipotent
characters.
2. The groups and some subgroups
2.1. The groups. We begin by fixing some notation which will be used throughout
our paper. Let q be a power of a prime p, and let Fq denote the finite field with q
elements. Also, m is a positive integer and n := 2m. We consider the finite
symplectic group Spn (q) over the field Fq . For the computations we are going to
perform it is convenient to work with a specific natural matrix representation of
this group which we now introduce.
Matrices are usually, but not always, denoted by boldface lower case letters, the
transposed of a matrix a is written as at . We write Im for the identity matrix of
degree m, and Jm for the (m × m)-matrix with ones along the anti-diagonal and
zeroes otherwise. Finally, let J˜n be the matrix
0
Jm
˜
Jn :=
.
−Jm 0
With these conventions we define
G := Spn (q) := {x ∈ GLn (q) | xt J˜n x = J˜n }.
We sometimes refer to n as the degree of G. The Weyl group of G is of type C m .
2.2. The parabolic subgroup. We let P be the maximal parabolic subgroup of G
fixing a line (in the natural G-vector space Fnq ). Then the Levi subgroup L of P
is (up to a cyclic direct factor) a symplectic group of degree n − 2 (we adopt the
convention that Sp0 (q) = {1}). Occasionally we consider the analogous parabolic
subgroup of L, and so on, in a recursive fashion. We therefore indicate the degree
of the ambient symplectic group by a subscript.
We now describe P through its Levi decomposition. First of all, for v ∈ Fqn−2
and z ∈ Fq we put


1 vt J˜n−2 z
un (v, z) :=  0
In−2
v .
0
0
1
Then, we let Un ≤ G be the subgroup
Un := {un (v, z) | v ∈ Fqn−2 , z ∈ Fq }.
Next, for x ∈ Spn−2 (q) and a ∈ F∗q , we write


a 0
0
0 ,
(1)
sn (x, a) :=  0 x
0 0 a−1
2
and put
Ln := sn (x, a) | x ∈ Spn−2 (q), a ∈ F∗q .
Then P := Pn := Un Ln is a maximal parabolic subgroup of G = Spn (q) fixing
the line h[1, 0, . . . , 0]t i. The group Un is the unipotent radical of P and Ln is its
Levi complement. We write L0n for the subgroup of Ln consisting of the matrices
sn (x, a) of (1) with a = 1. Then L0n ∼
= Spn−2 (q). (Moreover, L0n is the commutator
subgroup of Ln except if n = 4 and q = 2, 3 or n = 6 and q = 2.) Also, Ln = L0n ×A,
where A = {sn (In−2 , a) | a ∈ F∗q }. We finally put Pn0 := Un L0n .
If n = 2 or q is even, then Un is abelian. If n > 2 and q is odd, then Un is a special
p-group of structure q 1+(n−2) . In this case, the center Z(Un ) of Un is isomorphic to
the additive group of Fq and consists of the matrices un (0, z), z ∈ Fq , the central
quotient Un /Z(Un ) is isomorphic to the natural vector space Fqn−2 , and the action
of L0n on Un /Z(Un ) by conjugation is equivalent to the natural action of Spn−2 (q)
on Fqn−2 .
2.3. The characters of Pn . Assume that n ≥ 2, and put P := Pn , U := Un ,
L := Ln , L0 := L0n and P 0 := U L0 . The ordinary irreducible characters of P can be
classified into three types:
Type 1: Characters with U in their kernel.
Type 2: Characters with Z(U ) but not U in their kernel.
Type 3: Characters with Z(U ) not in their kernel.
Note that for n = 2 there are no characters of Type 2 (since Z(U ) = U in this
case). Using Clifford theory, we are now going to describe the characters of the
individual types in more detail and introduce a convenient notation.
2.3.1. Characters of Type 1. These are in bijection with Irr(L) via inflation. We
write 1ψ σ for the character of Type 1 corresponding to σ ∈ Irr(L). Thus
ψ σ = InflP
L (σ).
1
We have 1ψ σ (1) = σ(1).
2.3.2. Characters of Type 2. These are parametrized by Irr(Pn−2 ) in the following
way. A character of Type 2 is a character of the quotient P/Z(U ) which is a semidirect product of L with U/Z(U ). Since L0 is transitive on the non-zero vectors of
its natural module, there is just one L0 -orbit on the set of non-trivial characters of
U/Z(U ). The irreducible characters of Type 2 thus correspond to the irreducible
characters of the stabilizer in L of a non-trivial irreducible character of U/Z(U ).
Again we need a convenient choice for the purpose of computations.
Suppose that n ≥ 4 and let ξ be a nontrivial irreducible complex character
of the additive group of Fq . Define λ ∈ Irr(U ) by λ(un (v, z)) := ξ(v1 ), where
v = [v1 , . . . , vn−2 ]t ∈ Fqn−2 . Let Tn := TP (λ) denote the inertia subgroup of λ
in P . It is readily checked that Tn = Un P̃n−2 with P̃n−2 = Ũn−2 L̃n−2 , where
the subgroups Ũn−2 and L̃n−2 are embedded into L in the following way (missing
entries in the matrices below are zeroes):



 1

 | v ∈ Fqn−4 , z ∈ Fq ,
un−2 (v, z)t
Ũn−2 = 


1
3
and
L̃n−2
 −1
a






= 














a 0
0
0 x
0
0 0 a−1


 | x ∈ Spn−4 (q), a ∈ F∗q .







a
Clearly, P̃n−2 ∼
P
,
and
thus
there
is
a
bijection
between Irr(P̃n−2 ) and Irr(Pn−2 ).
= n−2
We construct a particular such bijection to describe the characters of Type 2. First,
∗
∗
we put Pn−2
:= {sn ((x−1 )t , 1) | x ∈ Pn−2 }. Then Pn−2 ∼
, and we write
= Pn−2
µ 7→ µ∗ for the corresponding bijection of irreducible characters. We then have
∗
Pn−2
× A = P̃n−2 × A (recall that A consists of the matrices sn (In−2 , a) for a ∈ F∗q ).
P∗
×A
Hence the map Irr(Pn−2 ) → Irr(P̃n−2 ), µ 7→ µ̃, defined by ResP̃n−2 (µ∗ ×1A ) = µ̃,
n−2
is a bijection.
Since λ is linear and Tn is the semi-direct product of U with P̃n−2 , it follows
that λ extends to an irreducible character λ̂ of Tn such that ResTP̃n (λ̂) = 1P̃n−2 .
n−2
Thus if µ is an irreducible character of Pn−2 , the induced character
Tn
n
ψ µ := IndP
Tn (λ̂ · InflP̃
2
n−2
(µ̃))
is an irreducible character of P of Type 2. Moreover, every irreducible character
of P of Type 2 arises this way. We have 2ψ µ (1) = (q n−2 − 1)µ(1).
2.3.3. Characters of Type 3. The description of these characters depends on the
parity of q, and we only describe them in the case of odd q. Here, the non-trivial
irreducible characters of Z(U ) fall into two orbits under the action of A. Let ζ1 and
ζ2 be representatives of the two orbits. There are unique irreducible characters ρ i
m−1
of U with ρi (1) = q m−1 (recall that n = 2m) such that ResU
ζi ,
Z(U ) (ρi ) = q
i = 1, 2.
Since L0 = Spn−2 (q) acts trivially on Z(U ), the ρi are invariant under L0 . Trivially, they are also invariant under Z = Z(G) = h−In i, which is contained in P
but not in P 0 . Since every element of P that fixes ρi also fixes ζi , it follows that
P 0 Z = P 0 × Z is the full stabilizer of ρi in P , i = 1, 2. Now the ρi extend to
characters ρ̂i of P 0 = U L0 (see [5, Theorem 2.4]). We denote the trivial extensions
of the ρ̂i to P 0 Z by the same symbols.
Every irreducible character ϑ of L0 ∼
= Spn−2 (q) has two extensions to L0 Z =
−
L0 × Z, namely ϑ · 1εZ with the sign ε ∈ {+, −}, where 1+
Z and 1Z denote the trivial
and non-trivial irreducible characters of Z, respectively. For each such ϑ we thus
get four irreducible characters of P of Type 3:
3 i,ε
ψϑ
0
P Z
ε
:= IndP
P 0 Z (ρ̂i · InflL0 Z (ϑ · 1Z )),
i = 1, 2 and ε ∈ {+, −}. Every character of Type 3 arises this way. We have
3 i,ε
ψ ϑ (1) = q m−1 (q − 1)ϑ(1)/2.
3. Restricting the Steinberg character
3.1. The restriction. For every standard Levi subgroup M of G, we denote the
Steinberg character of M by StM . We want to compute the restriction ResG
P (StG ).
P
(St
)
(see
e.g.,
[1,
Proposition
6.3.3]);
(St
)
=
Ind
It is well known that ResG
L
G
L
P
here L is the standard Levi complement Ln−2 of P = Pn . We thus have to compute
4
IndP
L (StL ). We can do this more generally by replacing StL by irreducible characters
of L0 = Spn−2 (q), extended trivially to L = L0 × (q − 1). Notice that StL is the
trivial extension of StL0 to L.
In computing the multiplicities of characters of Type 3 in such induced characters
for odd q, the Weil characters of L0 play a crucial role. Recall that ρ̂i denotes an
0
extension of ρi , an irreducible character of Un−2 , to Pn−2
= Un−2 L0n−2 . If q is odd
P0
and n ≥ 4, the characters ωi := ResLn−2
(ρ̂i ), i = 1, 2 are called the Weil characters
0
n−2
0 ∼
of L = Spn−2 (q). Gérardin has shown in [5, Corollary 4.4], that ωi is the sum of two
irreducible characters of degrees (q (n−2)/2 − 1)/2 and (q (n−2)/2 + 1)/2, respectively.
If H is a finite group, we write h−, −iH for the usual inner product on the set
of class functions on H.
Proposition 3.2. Suppose that q is odd and that n ≥ 4. Let σ be an irreducible
character of L0 = L0n−2 ∼
= Spn−2 (q). Denote the trivial extension of σ to L = L0 ×A
by σ as well. Then we have
IndP
L (σ) =
+
1
ψσ
X
µ∈Irr(Pn−2 )
+
D
0
ResL
Pn−2 (σ), µ
E
1,+
X
hσ · ω1 , ϑiL0 3ψ ϑ
2
Pn−2
ψµ
2,+
+ hσ · ω2 , ϑiL0 3ψ ϑ .
ϑ∈Irr(L0 )
Proof. Let us begin with characters of Type 1. Let τ ∈ Irr(L). We have
D
E
D
E
P
1
=
σ, ResP
IndP
L (InflL (τ ))
L (σ), ψ τ
L
P
=
hσ, τ iL ,
and thus 1ψ σ is the only constituent of IndP
L (σ) of Type 1.
Let us now consider characters of Type 2, and let µ ∈ Irr(Pn−2 ). In the following
computation we use Mackey’s theorem together with the fact that Tn L = P and
that L ∩ Tn = P̃n−2 . We have
E
D
2
2
= σ, ResP
(σ),
ψ
IndP
µ
L ( ψµ ) L
L
P
D
E
P
Tn
=
σ, ResP
L (IndTn (λ̂ · InflP̃n−2 (µ̃)))
L
D
E
L
Tn
Tn
=
σ, IndL∩Tn (ResL∩Tn (λ̂ · InflP̃ (µ̃)))
n−2
D
EL
Tn
Tn
L
=
ResP̃n−2 (σ), ResP̃ (λ̂ · InflP̃ (µ̃)))
n−2
n−2
P̃n−2
D
E
L
=
ResP̃n−2 (σ), µ̃
.
P̃n−2
Recall that L = L0 × A and σ = σ × 1A . Thus
0
×A
ResL
Pn−2 ×A (σ × 1A ) =
=
0
ResL
Pn−2 (σ) × 1A
D
E
X
0
ResL
Pn−2 (σ), µ (µ × 1A ).
µ∈Irr(Pn−2 )
P
Now Pn−2 × A = P̃n−2 × A, and ResP̃n−2
n−2
for characters of Type 2.
5
×A
(µ × 1A ) = µ̃, which proves the claim
Finally, let ϑ ∈ Irr(L0n−2 ), let i ∈ {1, 2} and ε ∈ {+, −}. We shall use the fact
that P 0 ZL = P and that L ∩ P 0 Z = L0 Z. We have
D
D
E
E
P
P 0Z
P
3 i,ε
ε
IndP
(σ),
ψ
=
σ,
Res
(Ind
(ρ̂
·
Infl
(ϑ
·
1
)))
0
0
L
ϑ
L
P Z i
LZ
Z
L
P
D
E
0
0
L
P Z
P Z
=
σ, IndL∩P 0 Z (ResL∩P 0 Z (ρ̂i · InflL0 Z (ϑ · 1εZ )))
L
D
E
P 0Z
ε
=
ResL
(σ),
Res
(ρ̂
)
·
(ϑ
·
1
)
i
L0 Z
L0 Z
Z
L0 ×Z
D
E
0
P
ε
=
σ, ResL0 (ρ̂i ) · ϑ 0 h1Z , 1Z iZ
L
= hσ, ωi · ϑiL0 h1Z , 1εZ iZ
= hσ · ω̄i , ϑiL0 h1Z , 1εZ iZ ,
i,ε
where ω̄i denotes the complex conjugate of ωi . So the character 3ψ ϑ only occurs in
IndP
L (σ) if ε = +, and then it occurs with multiplicity (σ · ω̄i , ϑ). By construction
of the Weil characters, ω̄i is again a Weil character, and the result follows.
The above description easily extends to the case of n = 2. In this case L = A and
L0 is the trivial group. Also, P/Z is a Frobenius group of order q(q − 1)/2. Thus P
has two irreducible characters of degree (q − 1)/2 which have Z in their kernel.
2,+
i,+
3 1,+
+ 3ψ 1 .
These are the characters named 3ψ 1 , i = 1, 2, and 1P
L = 1P + ψ 1
Corollary 3.3. Suppose that q is odd and that n ≥ 4. Then we have
ResG
P (StG ) =
+
1
ψ StL
X
µ∈Irr(Pn−2 )
+
X
D
0
ResL
Pn−2 (StL0 ), µ
1,+
hStL0 · ω1 , ϑiL0 3ψ ϑ
E
2
Pn−2
ψµ
2,+
+ hStL0 · ω2 , ϑiL0 3ψ ϑ .
ϑ∈Irr(L0 )
This shows in particular that the constituents of Type 2 of ResG
P (StG ) can be found
recursively.
3.4. Constituents of Type 3. To find the multiplicities of the Type 3 characters
in the restriction of StG to P , we have to compute the multiplicities of the irreducible
constituents of the product StL0 ·ωi for i = 1, 2. It follows from the work of Gérardin
(see [5, Corollary 4.8.1]) that ω1 and ω2 agree on semisimple elements. On the other
hand, the Steinberg character vanishes on elements which are not semisimple. Hence
StL0 · ω1 = StL0 · ω2 .
To simplify notation we discuss the analogous product of characters in G. Thus
let St := StG and let ω be one of the Weil characters of G. The class function
St · ω vanishes except on semisimple elements. Hence it is a linear combination of
Deligne-Lusztig generalized characters RT,ϑ , where T runs through the maximal tori
of G and ϑ through the irreducible characters of T (see [1, Corollary 7.5.7]). The
Deligne-Lusztig characters RT,ϑ are either equal or orthogonal (see [3, Corollary
4.5.5]). Hence in order to write St · ω as a linear combination of the RT,ϑ , it suffices
to compute the scalar products hSt · ω, RT,ϑ iG = hω, St · RT,ϑ iG .
6
By [1, Proposition 7.5.4], we have St · εG εT RT,ϑ = IndG
T (ϑ), where εG and εT
are signs as in [1, p. 199]. Therefore,
hSt · ω, εG εT RT,ϑ iG
= hω, St · εG εT RT,ϑ iG
D
E
=
ω, IndG
T (ϑ)
G
G
= ResT (ω), ϑ T .
Through the work of Lusztig, the decomposition of the RT,ϑ into irreducible characters can be computed. This will be used in the following section to write the
tensor product St ⊗ ω as a sum of irreducible characters in the case of G = Sp 4 (q).
3.5. An example. To illustrate Corollary 3.3, we compute the restriction of the
Steinberg character of Sp4 (q), q odd, to the maximal parabolic subgroup P4 .
Let q be odd, G = Sp4 (q), P = P4 , and L = L4 = L0 × A. We have L0 ∼
=
Sp2 (q) ∼
= SL2 (q) and its character table can easily be found in the literature (e.g.,
in [14, p. 128]).
(a) We first determine the decomposition of the product of the Weil characters
ωi , i = 1, 2 of L0 with its Steinberg character St:
X
St · ω1 = St · ω2 =
ϑ.
ϑ ∈ Irr(SL2 (q))
ϑ(1) 6= 1, (q − 1)/2
This can easily be checked with the character table of SL2 (q). The decomposition
has been found with the help of CHEVIE [4].
(b) We next determine the restriction of StL0 to P2 . By the remarks following
0
Proposition 3.2 we have ResL
(StL0 ) = 1 + µ1 + µ2 , where the two characters µi
P̃2
have degree (q − 1)/2.
(c) From Corollary 3.3 we obtain
P
ResG
P (StG ) = IndL (StL ) =
1
+
2
+
ψ StL
ψ 1 + 2ψ µ1 + 2ψ µ2
X
3 1,+
ψϑ
2,+
+ 3ψ ϑ .
ϑ ∈ Irr(SL2 (q))
ϑ(1) 6= 1, (q − 1)/2
In particular, ResG
P (StG ) is multiplicity free.
4. Decomposition of the character St · ω in Sp4 (q)
Let q be an odd prime power and let ω be one of the two Weil characters of
G = Sp4 (q). In this section we will show that the product StG ·ω is multiplicity-free.
By Corollary 3.3 and the result of Subsection 3.5, this implies that for G = Sp 6 (q),
the restriction of StG to P is multiplicity free as well.
Let G = Sp4 (q). In order to decompose St · ω, we shall use the method sketched
in Subsection 3.4. The character values of ω are given explicitly in [15], and our
first proof of the
following lemma was obtained by a direct computation of the
scalar products ResG
T (ω), ϑ T . We are very much indebted to Alex Zalesskii who
suggested a more conceptual method for computing these.
7
Let T be a maximal torus of G, so that
T ∈ {Zq2 +1 , Zq2 −1 , Zq−1 × Zq−1 , Zq+1 × Zq+1 , Zq+1 × Zq−1 }.
In the notation of [15], Zq2 +1 = hζi, Zq2 −1 = hθi, Zq−1 = hγi and Zq+1 = hηi.
Given x ∈ {ζ, θ, γ, η}, let x̂ be the character such that x̂(x) = x̃, where x̃ ∈ C is
defined in [15].
Lemma 4.1. Let T be a maximal
torus and ϑ ∈ Irr(T ). Then the non-zero values
of the inner product ResG
T (ω), ϑ T are as given in Table 1.
ResG
T (ω), ϑ
1
2
4
T
(T, ϑ)
Conditions
k
(Zq2 +1 , ζ̂ )
(Zq2 −1 , θ̂ k )
(Zq−1 × Zq−1 , γ̂ k × γ̂ ` )
(Zq+1 × Zq+1 , η̂ k × η̂ ` )
(Zq+1 × Zq−1 , η̂ k × γ̂ ` )
(Zq2 −1 , θ̂ k )
(Zq−1 × Zq−1 , γ̂ k × γ̂ ` )
(Zq+1 × Zq−1 , η̂ k × γ̂ ` )
(Zq−1 × Zq−1 , γ̂ k × γ̂ ` )
2
k=
6 q 2+1
2
k 6= q 2−1
k 6= q−1
and ` 6= q−1
2
2
q+1
k 6= 2 and ` 6= q+1
2
q−1
k 6= q+1
2 and ` 6= 2
2
k = q 2−1
q−1
either k = q−1
2 or ` = 2
q+1
q−1
k 6= 2 and ` = 2
k = ` = q−1
2
Table 1
Proof. Let us denote the regular character of a finite group H by regH . If C
is a cyclic group of even order, we denote by αC its unique non-trivial irreducible
complex character with values in {±1}.
The following two facts about the restriction of ω to the maximal tori of G can
be derived from [5, Corollary 4.8.1]. If T = Zq2 +1 , then ResG
T (ω) = regT − αT .
(ω)
=
reg
+
α
.
It
follows
from
[5,
Corollary 2.5], that
If T = Zq2 −1 , then ResG
T
T
T
ResG
(ω)
=
µ
⊗
ν,
where
Sp
(q)
×
Sp
(q)
is
embedded
naturally into G
2
2
Sp2 (q)×Sp2 (q)
through an orthogonal decomposition of the underlying symplectic space, and where
µ and ν denote Weil characters of Sp2 (q). These remarks easily give the result of
Table 1.
Proposition 4.2. In the notation of [15], let Ω be the subset of Irr(G) consisting
of the characters listed in Table 2. Then hSt · ω, χiG = 1 or 0 according as χ ∈ Ω
or χ ∈ Irr(G) \ Ω.
Proof. This follows from the considerations in Subsection 3.4, Lemma 4.1, and
the relations between the irreducible characters and the RT,ϑ given in [15, Sections
5–7].
It is clear that the methods used in this section can be generalized to obtain information about the decomposition of StG · ω for symplectic groups G = Spn (q) of
arbitrary degree n.
5. An application to the decomposition numbers of Sp6 (q)
Let ` be a prime number bigger than 3. In this section we are interested in the
`-modular representations of the groups Sp6 (q) for odd prime powers q prime to `.
8
Character
Parameter
Character
Parameter
χ1 (k)
χ3 (i, j)
−χ5 (k, `)
χ9 (i)
ξ30 (i)
−ξ22 (j)
ξ42 (i)
Φ8
θ1
θ13
k ∈ R1
i, j ∈ T1 , i 6= j
k ∈ T 2 , ` ∈ T1
i ∈ T1
i ∈ T1
j ∈ T2
i ∈ T1
−χ2 (`)
χ4 (i, j)
χ7 (j)
−ξ10 (j)
−ξ21 (j)
ξ41 (i)
Φ7
Φ9
θ2
` ∈ R2
i, j ∈ T2 , i 6= j
j ∈ T2
j ∈ T2
j ∈ T2
i ∈ T1
Table 2
Our main result is a proof of a version of Donovan’s conjecture for this class of
groups.
Theorem 5.1. Let ` > 3 be a prime number and F an algebraically closed field of
characteristic `. Fix a non-negative integer d. Then there are only finitely many
Morita equivalence classes among the unipotent `-blocks of defect d of the group
algebras F Sp6 (q), where q runs through the odd prime powers not divisible by `.
Proof. It suffices to consider prime powers q such that ` divides
|Sp6 (q)| = q 9 (q − 1)3 (q + 1)3 (q 2 + q + 1)(q 2 + 1)(q 2 − q + 1).
Since we assume that q is not divisible by `, and ` > 3, it follows that ` divides
exactly one of q − 1, q + 1, q 2 + q + 1, q 2 + 1 or q 2 − q + 1. In the latter three cases,
the defect groups of the blocks considered are cyclic (see [2]), and therefore there
are only finitely many of them of a given defect (up to Morita equivalence).
Since ` > 3, it does not divide the order of the Weyl group of Sp6 (q). By a result
of Puig [12], the unipotent `-blocks of the groups Sp6 (q), where q runs through the
prime powers with ` | q − 1, are all Morita equivalent to each other.
We may therefore restrict attention to prime powers q such that ` | q + 1.
Moreover, by [7, Propositions 9.1, 9.2], it suffices to show that the Cartan invariants
of such blocks are bounded by a function in d. By Brauer reciprocity and the fact
that the defect of a block bounds its number of modular irreducible characters,
we may instead bound the decomposition numbers in terms of d. Since the defect
groups of the blocks in question are abelian, it suffices, by [8, Proposition 4.5], to
show that the decomposition numbers of the unipotent characters in such a block
are bounded by a function in the defect.
Let q be a prime power such that ` | q +1. The group G := Sp6 (q) has two unipotent characters lying in the Sp4 (q)-Harish-Chandra-series, labelled by C20 and C200 .
The other unipotent characters lie in the principal series and hence are labelled by
bi-partitions of 6. The Steinberg character is labelled by (−, 13 ) (see, e.g., [1, 13.7,
13.8]). By the results of Fong and Srinivasan [2], G has two unipotent `-blocks, one
of cyclic defect containing the unipotent characters with labels (21, −) and (−, 21),
and the principal block. Table 3 is an approximation to the decomposition matrix
9
of the unipotent characters of the principal block of G. The first column of that
table gives the labels of the unipotent characters. The other columns correspond to
projective F G-modules, whose ordinary characters are denoted by Φ1 , . . . , Φ10 . An
entry of Table 3 in a row corresponding to the unipotent character χ and a column
corresponding to the projective character Φ is the inner product hχ, Φi. Zeroes are
not printed. (This table is an approximation to the decomposition matrix in the
sense that we make no statements about the indecomposability of the Φi .)
Let us shortly describe the origin of these projective characters. Let P denote
the parabolic subgroup of G considered in Section 2.2 with Levi subgroup L ∼
=
Sp4 (q) × GL1 (q). The principal `-block contains five simple F L-modules, and there
is one unipotent character of `-defect 0. The decomposition matrix of L has been
determined by Okuyma and Waki [11]. Harish-Chandra inducing the six projective
indecomposable characters lying in unipotent blocks of L, we obtain the projective
characters Φ1 , . . . , Φ4 , Φ6 , and Φ7 . These, and the following Harish-Chandra
induced characters have been computed with the CHEVIE share package of GAP3,
developed by Geck, Lübeck, and Michel (see [13]).
Let Ψ1 denote the character of the projective cover of the trivial F G-module.
Since Φ1 is obtained by inducing the projective cover of the trivial F P -module
to G, it follows that Ψ1 is contained in Φ1 (in the sense that Φ1 − Ψ1 is a proper
character). Moreover, Φ1 has only unipotent constituents (being contained in the
permutation character of G on the Borel subgroup of G). The permutation character of G on P equals (3, −) + (2, 1) + (21, −) (we identify a unipotent character
with its label). Since (21, −) is not contained in the principal `-block of G, and
since ` divides the index of P in G, it follows that the reduction of (2, 1) modulo ` contains a trivial constituent. This implies that (2, 1) is a constituent of Ψ1 .
The Steinberg character (−, 13 ) is also contained in Ψ1 (see, e.g., [6]). Using the
degrees of its constituents, it is now easy to check that Ψ1 = Φ1 . (Similarly, Φ2 ,
being Harish-Chandra induced from the defect 0 unipotent character of L, has no
non-unipotent constituents and is the character of an indecomposable projective
F G-module. Using the `-modular representation theory of the Iwahori-Hecke algebra of type C3 with parameters −1, Jürgen Müller has given a more elegant proof
of the indecomposability of Φ1 and Φ2 .)
The projective character Φ5 is the Harish-Chandra induced Gelfand-Graev character of the Levi-subgroup Sp2 (q)×GL2 (q), and Φ10 is the Gelfand-Graev character
of G (restricted to the principal block). It is easy to see that Ψ = (21, −) + (−, 21)
is a projective character in the non-principal unipotent block of G. The product
Ψ · (21, −) contains the character of the projective cover of the trivial F G-module,
which was shown above to be equal to Φ1 , and Φ8 = Ψ · (21, −) − Φ1 . Finally, the
product Φ2 · C20 yields Φ9 . The products of these characters have been computed
with the Maple-based CHEVIE system (see [4]), containing in particular the table
of unipotent characters of Sp6 (q) computed by Lübeck (see [10]).
Corollary 3.3 and Proposition 4.2 show that the restriction of StG to P is multiplicity free. By the results of Okuyama and Waki [11] on the decomposition
numbers of Sp4 (q), and by Clifford theory, the decomposition numbers in P are
bounded in terms of d, independently of q. This in turn implies that the decomposition numbers occurring in StG are bounded in terms of d, completing the proof.
10
(3, −)
(2, 1)
(−, 3)
C20
(1, 2)
(12 , 1)
(1, 12 )
(13 , −)
C200
(−, 13 )
Φ1
1
1
1
Φ2
1
1
1
1
1
1
1
Φ3
Φ4
Φ5
Φ6
Φ7
1
1
1
1
1
1
Φ8
Φ9
Φ10
1
1
1
1
1
1
2
1
1
2
1
1
1
1
1
1
(q + 1)/2 (q − 1)/2
1
Table 3. An approximation to the decomposition matrix of Sp6 (q)
Christoph Köhler has shown that in fact all the entries of Table 3 except (q ±
1)/2 are true decomposition numbers, but this stronger information is not needed
to prove the above theorem. Köhler’s results will appear as part of his PhDthesis [9]. It is also clear that the results of Corollary 3.3 and Proposition 4.2 can
be used to improve the known bounds on the two unknown decompositon numbers
considerably.
Table 3 is also an approximation to the 3-modular decomposition matrix of
G = Sp6 (q) if 3 | q + 1. In this case, however, the Sylow 3-subgroup of G is
not abelian, and we cannot appeal to [8, Proposition 4.5] to bound all decomposition numbers in the principal 3-block by the decomposition numbers occuring for
unipotent characters.
Acknowledgements
We thank Meinolf Geck, Frank Lübeck, and Gunter Malle for clarifying discussions on Deligne-Lusztig theory. We also thank Alex Zalesskii for his hints on the
restrictions of the Weil characters to maximal tori. These suggestions led to a
substantial simplification of some of our computations.
Most of this work was done during a visit of the second author at the Department
of Mathematics of the University of Auckland. He wishes to express his sincere
thanks to all the persons of the department for their hospitality, and also to the
Marsden Fund of New Zealand who supported his visit via grant #9144/336828.
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J.A.: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
G.H.: Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen,
Germany
E-mail address: J.A.: [email protected]
E-mail address: G.H.: [email protected]
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